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Lines of the second order

2
2
2
x 0 = 0, y 0 = 0, then the equation of a circle looks like: x + y = R . It is obvious, that
equation (7.2) is the equation of the second order relatively to variables x and y. Having opened
parentheses in (7.2), we will get the equation:
2
2
2
2
2
x + y − 2xx 0 − 2yy 0 + x + y − R = 0. (7.3)
0 0
On comparing equation (7.3) with general equation (7.1), the following conclusions can be
made: 1. there isn’t term with multiple xy; 2. coefficients at x and y are equal.
Obviously, coordinates of any point, which belongs to the circle, satisfy equation (7.3). It
is easy to prove that the reverse statement is true as well: if in general equation (7.1) 2B = 0
2
2
and A = C = 1 (it doesn’t conflict with general considerations), then the equation x + y +
2Dx + 2Ey + F = 0 describes the circle. It is enough to make a complete square relatively to
variables x and y and get the equation, similar to (7.2).

7.2. Ellipse

Definition 7.2. An ellipse is a geometric place of points on a plane, the sum of
distances from two fixed points, that are called focuses, is constant and is more
than the distance between focuses. Let’s mark F 1 and F 2 — focuses of the ellipse,
2c = |F 1 F 2 | — the distance between focuses, 2a — the sum of distances from an
arbitrary point of the ellipse to focuses. ✓

According to the definition, 2a > 2c or a > c.
Let’s take a rectangular system of coordinates in such a way, that focuses of an ellipse are
located on x-axis and the beginning of coordinated halves the interval F 1 F 2 (fig. 7.2). Then
coordinates of focuses are: F 1 (−c, 0), F 2 (c, 0). Let’s get the equation of an ellipse in the chosen

system of coordinates. Suppose, M(x, y) is an arbitrary point on a plane. Let’s mark τ 1 and τ 2
— distances from point M to focuses: τ 1 = |F 1 M| , τ 2 = |F 2 M| . Intervals τ 1 and τ 2 are called
as focal radii of point M. As point M belongs to the ellipse, then according to the definition,

τ 1 + τ 2 = 2a. (7.4)
Let’s evaluate τ 1 and τ 2 using the formula of a distance between two points:

√ √
2
τ 1 = (x + c) + y , τ 2 = (x − c) + y . (7.5)
2
2
2
y

M(x, y)
τ 1 τ 2

O . . . . . . . . .
F 1 (−c, 0) F 2 (c, 0) x

Figure 7.2 – Construction of ellipse

Having substituted in (7.4), we will get:
√ √
2
2
2
2
(x + c) + y + (x − c) + y = 2a. (7.6)
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